Ultralimit Explained

In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces

Xn

. The concept captures the limiting behavior of finite configurations in the

Xn

spaces employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure convergence. Ultralimits generalize Gromov–Hausdorff convergence in metric spaces.

Ultrafilters

An ultrafilter, denoted as ω, on the set of natural numbers is a set of nonempty subsets of (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and also which, given any subset X of, contains either X or . An ultrafilter on is non-principal if it contains no finite set.

Limit of a sequence of points with respect to an ultrafilter

In the following, ω is a non-principal ultrafilter on

N

.

If

(xn)n\in

is a sequence of points in a metric space (X,d) and xX, then the point x is called ω-limit of xn, denoted as

x=\lim\omegaxn

, if for every

\epsilon>0

it holds that

\{n:d(xn,x)\le\epsilon\}\in\omega.

It is observed that,

x=\limn\toinftyxn

in the standard sense,

x=\lim\omegaxn

. (For this property to hold, it is crucial that the ultrafilter should be non-principal.)

A fundamental fact states that, if (X,d) is compact and ω is a non-principal Ultrafilter on

N

, the ω-limit of any sequence of points in X exists (and is necessarily unique).

In particular, any bounded sequence of real numbers has a well-defined ω-limit in

R

, as closed intervals are compact.

Ultralimit of metric spaces with specified base-points

Let ω be a non-principal ultrafilter on

N

. Let (Xn,dn) be a sequence of metric spaces with specified base-points pnXn.

Suppose that a sequence

(xn)n\inN

, where xnXn, is admissible. If the sequence of real numbers (dn(xn,pn))n is bounded, that is, if there exists a positive real number C such that

dn(xn,pn)\leC

, then denote the set of all admissible sequences by

lA

.

It follows from the triangle inequality that for any two admissible sequences

x=(xn)n\inN

and

y=(yn)n\inN

the sequence (dn(xn,yn))n is bounded and hence there exists an ω-limit

\hatdinfty(x,y):=\lim\omegadn(xn,yn)

. One can define a relation

\sim

on the set

lA

of all admissible sequences as follows. For

x,y\inlA

, there is

x\simy

whenever

\hatdinfty(x,y)=0.

This helps to show that

\sim

is an equivalence relation on

lA.

The ultralimit with respect to ω of the sequence (Xn,dn, pn) is a metric space

(Xinfty,dinfty)

defined as follows.[1]

Written as a set,

Xinfty=lA/{\sim}

.

For two

\sim

-equivalence classes

[x],[y]

of admissible sequences

x=(xn)n\inN

and

y=(yn)n\inN

, there is

dinfty([x],[y]):=\hatdinfty(x,y)=\lim\omegadn(xn,yn).

This shows that

dinfty

is well-defined and that it is a metric on the set

Xinfty

.

Denote

(Xinfty,dinfty)=\lim\omega(Xn,dn,pn)

.

On base points in the case of uniformly bounded spaces

Suppose that (Xn,dn) is a sequence of metric spaces of uniformly bounded diameter, that is, there exists a real number C > 0 such that diam(Xn) ≤ C for every

n\inN

. Then for any choice pn of base-points in Xn every sequence

(xn)n,xn\inXn

is admissible. Therefore, in this situation the choice of base-points does not have to be specified when defining an ultralimit, and the ultralimit

(Xinfty,dinfty)

depends only on (Xn,dn) and on ω but does not depend on the choice of a base-point sequence

pn\inXn

. In this case one writes

(Xinfty,dinfty)=\lim\omega(Xn,dn)

.

Basic properties of ultralimits

  1. If (Xn,dn) are geodesic metric spaces then

(Xinfty,dinfty)=\lim\omega(Xn,dn,pn)

is also a geodesic metric space.
  1. If (Xn,dn) are complete metric spaces then

(Xinfty,dinfty)=\lim\omega(Xn,dn,pn)

is also a complete metric space.[2] [3] Actually, by construction, the limit space is always complete, even when (Xn,dn)is a repeating sequence of a space (X,d) which is not complete.[4]
  1. If (Xn,dn) are compact metric spaces that converge to a compact metric space (X,d) in the Gromov–Hausdorff sense (this automatically implies that the spaces (Xn,dn) have uniformly bounded diameter), then the ultralimit

(Xinfty,dinfty)=\lim\omega(Xn,dn)

is isometric to (X,d).
  1. Suppose that (Xn,dn) are proper metric spaces and that

pn\inXn

are base-points such that the pointed sequence (Xn,dn,pn) converges to a proper metric space (X,d) in the Gromov–Hausdorff sense. Then the ultralimit

(Xinfty,dinfty)=\lim\omega(Xn,dn,pn)

is isometric to (X,d).
  1. Let κ≤0 and let (Xn,dn) be a sequence of CAT(κ)-metric spaces. Then the ultralimit

(Xinfty,dinfty)=\lim\omega(Xn,dn,pn)

is also a CAT(κ)-space.[5]
  1. Let (Xn,dn) be a sequence of CAT(κn)-metric spaces where

\limn\toinfty\kappan=-infty.

Then the ultralimit

(Xinfty,dinfty)=\lim\omega(Xn,dn,pn)

is real tree.[5]

Asymptotic cones

An important class of ultralimits are the so-called asymptotic cones of metric spaces. Let (X,d) be a metric space, let ω be a non-principal ultrafilter on

N

and let pn ∈ X be a sequence of base-points. Then the ω - ultralimit of the sequence

(X,

d
n

,pn)

is called the asymptotic cone of X with respect to ω and

(pn)n

and is denoted

Cone\omega(X,d,(pn)n)

. One often takes the base-point sequence to be constant, pn = p for some p ∈ X; in this case the asymptotic cone does not depend on the choice of p ∈ X and is denoted by

Cone\omega(X,d)

or just

Cone\omega(X)

.

The notion of an asymptotic cone plays an important role in geometric group theory since asymptotic cones (or, more precisely, their topological types and bi-Lipschitz types) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular.[6] Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.[7]

Examples

  1. Let (X,d) be a compact metric space and put (Xn,dn)=(X,d) for every

n\inN

. Then the ultralimit

(Xinfty,dinfty)=\lim\omega(Xn,dn)

is isometric to (X,d).
  1. Let (X,dX) and (Y,dY) be two distinct compact metric spaces and let (Xn,dn) be a sequence of metric spaces such that for each n either (Xn,dn)=(X,dX) or (Xn,dn)=(Y,dY). Let

A1=\{n|(Xn,dn)=(X,dX)\}

and

A2=\{n|(Xn,dn)=(Y,dY)\}

. Thus A1, A2 are disjoint and

A1\cupA2=N.

Therefore, one of A1, A2 has ω-measure 1 and the other has ω-measure 0. Hence

\lim\omega(Xn,dn)

is isometric to (X,dX) if ω(A1)=1 and

\lim\omega(Xn,dn)

is isometric to (Y,dY) if ω(A2)=1. This shows that the ultralimit can depend on the choice of an ultrafilter ω.
  1. Let (M,g) be a compact connected Riemannian manifold of dimension m, where g is a Riemannian metric on M. Let d be the metric on M corresponding to g, so that (M,d) is a geodesic metric space. Choose a base point pM. Then the ultralimit (and even the ordinary Gromov-Hausdorff limit)

\lim\omega(M,nd,p)

is isometric to the tangent space TpM of M at p with the distance function on TpM given by the inner product g(p). Therefore, the ultralimit

\lim\omega(M,nd,p)

is isometric to the Euclidean space

Rm

with the standard Euclidean metric.[8]
  1. Let

(Rm,d)

be the standard m-dimensional Euclidean space with the standard Euclidean metric. Then the asymptotic cone
m,
Cone
\omega(R

d)

is isometric to

(Rm,d)

.
  1. Let

(Z2,d)

be the 2-dimensional integer lattice where the distance between two lattice points is given by the length of the shortest edge-path between them in the grid. Then the asymptotic cone
2,
Cone
\omega(Z

d)

is isometric to

(R2,d1)

where

d1

is the Taxicab metric (or L1-metric) on

R2

.
  1. Let (X,d) be a δ-hyperbolic geodesic metric space for some δ≥0. Then the asymptotic cone

Cone\omega(X)

is a real tree.[5] [9]
  1. Let (X,d) be a metric space of finite diameter. Then the asymptotic cone

Cone\omega(X)

is a single point.
  1. Let (X,d) be a CAT(0)-metric space. Then the asymptotic cone

Cone\omega(X)

is also a CAT(0)-space.[5]

References

See also

Notes and References

  1. John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ; Definition 7.19, p. 107.
  2. L.Van den Dries, A.J.Wilkie, On Gromov's theorem concerning groups of polynomial growth and elementary logic. Journal of Algebra, Vol. 89(1984), pp. 349 - 374.
  3. John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ; Proposition 7.20, p. 108.
  4. Bridson, Haefliger "Metric Spaces of Non-positive curvature" Lemma 5.53
  5. M. Kapovich B. Leeb. On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds, Geometric and Functional Analysis, Vol. 5 (1995), no. 3, pp. 582 - 603
  6. John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003.
  7. [Cornelia Druţu]
  8. Yu. Burago, M. Gromov, and G. Perel'man. A. D. Aleksandrov spaces with curvatures bounded below (in Russian), Uspekhi Matematicheskih Nauk vol. 47 (1992), pp. 3 - 51; translated in: Russian Math. Surveys vol. 47, no. 2 (1992), pp. 1 - 58
  9. John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ; Example 7.30, p. 118.